LAPLACE TRANSFORM SOLUTION of IVP's

Assume an IVP with constant coefficients is given is the form

$$a_0\frac{d^ny}{dt^n} + a_1\frac{d^{n-1}y}{dt^{n-1}} + \cdots + a_n y = g(t),$$

with $y(t_0)=y_0,\ y'(t_0) = y'_0,\ \ldots,\ y^{(n-1)}(t_0) = y^{(n-1)}_0.$

Basic Method:

• take Laplace transform of both sides of equation
  
  $${\cal L}\{a_0\frac{d^ny}{dt^n} + a_1\frac{d^{n-1}y}{dt^{n-1}} + \cdots + a_n y\} = {\cal L}\{g(t)\},$$
  
  

• use algebra to find ${\cal L}\{y(t)\} = Y(s)$,
• write $Y(s)$ as $Y(s) = Y_1(s) + Y_2(s) + \cdots + Y_m(s),$
• determine $y_i(t) = {\cal L}^{-1}\{Y_i(s)\}$, for $i = 1, 2, \ldots, m$,
• determine $y(t) = y_1(t) + y_2(t) + \cdots + y_m(t).$

The Unit Step Function

is defined by $u_b(t) = \left\{\begin{array}{ll}0&t<b\\ 1&t\geq b\end{array}\right.$.

Brief Table of Laplace Transforms

$y(t) = {\cal L}^{-1}\{Y(s)\}$	$Y(s) = {\cal L}\{y(t)\}$	$s$
$t^n$	$n!/s^{n+1}$	$s > 0$
$e^{at}t^n$	$n!/(s-a)^{n+1}$	$s > a$
$\sin(at)$ or $\cos(at)$	$(a\mbox{ or }s)/(s^2+a^2)$	$s > 0$
$\sinh(at)$ or $\cosh(at)$	$(a \mbox{ or } s)/(s^2-a^2)$	$s>\vert a\vert$
$e^{bt}y(t)$	$Y(s-b)$
$y^{(n)}(t)$	$s^nY(s)-s^{n-1}y(0)\cdots -y^{(n-1)}(0)$
$u_b(t)y(t-b)$	$e^{-bs}Y(s)$